53-] DETERMINATION OF CENTROIDS. $ l 



(25) A painter's palette is formed by cutting a small circle of radius 

 b out of a circular disc of radius a, the distance between the centres 

 being c. It is required to find the distance of the centroid of the 

 remainder from the centre of the larger circle. (Routh.) 



(26) The arch constructed of brick over a door is in the form of a 

 quadrant of a circular ring. The door is 5 ft. wide ; i-J- lengths of 

 brick are used (say 12 in.). Find the centroid of the arch. 



Find the co-ordinates of the centroid for the following plane areas : 



(27) Area bounded by the parabola y*=. ^ax y the axis of x, and the 

 ordinate y. 



(28) Area bounded by the curve jy = sin.# from x=o \.QX = TT 

 and the axis of x. 



(29) Quadrant of an ellipse. 



(30) Elliptic segment bounded by the chord joining the ends of 

 the major and minor axes. 



(31) Show, by Art. 28, that the centroid of the surface of a right 

 circular cone lies at a distance from the base equal to one-third of 

 the height. 



(32) Find the centroid of the portion of the surface of a right cir- 

 cular cone cut out by two planes through the axis inclined at an angle <. 



(33) Find the centroid of the area of the earth's surface contained 

 between the tropic of Cancer (latitude = 23 28') and the arctic circle 

 (polar distance = 23 28'). 



(34) Regarding the earth as a homogeneous sphere of density 

 10 = 5.5, how mucn would its centroid be displaced by superimposing 

 over the area bounded by the arctic circle an ice-cap of a uniform thick- 

 ness of 10 miles? 



(35) A bowl in the form of a hemisphere is closed by a circular lid 

 of a material whose density is three times that of the bowl. Find the 

 centroid. 



(36) Determine the centroid of a homogeneous solid hemisphere. 



(37) Find the centroid of a frustum of a cone, the radii of the 

 bases being r^ r 2 ; the height of the frustum, h. 



(38) Show that the formula for the frustum of the cone applies like- 

 wise to the frustum of any pyramid of the same height h if r lt r 2 are 

 any two homologous linear dimensions of the two bases. 



